You have to turn the key to unlock it then depress the button the key goes in, to open it. So it worked correctly it sounds like.

So as I understood correctly the key should be inside the lock when I push it inside. Is it correct? Also if car is centrally unlocked (by pulling the lock knob upward of driver side), even then do I need to unlock the trunk by turning the key anti-clock wise and push the lock inside or just put the key and push it inside is enough without turning the key to open the trunk?

--- But still the trunk will not open at first shot. So I am thinking to open the trunk lock and properly clean it and may be I would found some thing is broken inside.
I am looking some guidance and steps how to open trunk lock and service it properly.